I am using machine learning models to predict an ordinal variable (values: 1,2,3,4, and 5) using 7 different features. I posed this as a regression problem, so the final outputs of a model are continuous variables. So an evaluation box plot looks like this:
I experiment with both linear (linear regression, linear SVMs) and nonlinear models (SVMs with RBF, Random forest, Gradient boosting machines ). The models are trained using cross-validation (~1600 samples), and 25% of the dataset is used for testing (~540 samples). I am using R-squared and Root Mean Square Error (RSME) to evaluate the models on test samples. I am interested in finding an evaluation measure to compare linear models to nonlinear ones.
This is done for scientific research. It was pointed out that R-square might not be an appropriate measure for nonlinear models, and that the Chi-Square test would be a better measure for goodness of fit.
The problem is, I am not sure what is the best way to do it. When I browse Chi-square as the goodness of fit, I only get examples where the Chi-square test is used to see whether some categorical samples fit a theoretical expectation, such as here. So here are my considerations/questions:
One way I could think of is to categorize predicted (continuous) values into bins, and compare predicted distribution to the ground truth distribution using the Chi-Square test. But that doesn't make much sense, i.e. we have a machine learning model that perfectly predicts ground truth values 2,3, and 4, and values 5 predicts as 1, and values 1 as 5 - Chi-Square test that I propose here would reject the null hypothesis, although the model is mispredicting 2 out of 5 values.
As referred to in a tutorial from USC I could use formula (1) to compute Chi-Square value, where experimentally measured quantities (xi) are my ground truth values, and hypothesized values (mui) are my predicted values. My question is, what is the variance? If we observe each value 1,2,3,4, and 5 as a distinct category, then the variance of ground truth within each category is equaled to zero. Also, how one computes the degree of freedom (N-r)?
Related to the statement I am interested in finding an evaluation measure to compare linear models to nonlinear is the Chi-Square test the best (or even good) choice? What I've seen so far in machine learning competitions for regression tasks, either MSE or RSME are used for evaluation.